Diffusion and wave behaviour in linear Voigt model

TL;DR

This paper investigates the diffusion and wave behaviors in a third-order parabolic equation modeling dissipative media, providing asymptotic approximations and analyzing hyperbolic and parabolic dynamics using slow and fast time scales.

Contribution

It offers a rigorous asymptotic analysis of a third-order parabolic equation, elucidating the transition between wave and diffusion behaviors in various physical media.

Findings

Establishment of asymptotic approximations for solutions.

Analysis of hyperbolic and parabolic behavior using time scales.

Application to models of dissipative media.

Abstract

A boundary value problem related to a third- order parabolic equation with a small parameter is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third-order parabolic operator regularizes various non linear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution is estimated by means of slow time and fast time. As consequence, a rigorous asymptotic approximation for the solution is established.